Günther R. Raidl

Hybrid metaheuristics in combinatorial optimization: A survey

Research in metaheuristics for combinatorial optimization problems has lately experienced a noteworthy shift towards the hybridization of metaheuristics with other techniques for optimization. At the same time, the focus of research has changed from being rather algorithm-oriented to being more problem-oriented. Nowadays the focus is on solving the problem at hand in the best way possible, rather than promoting a certain metaheuristic. This has led to an enormously fruitful cross-fertilization of different areas of optimization. This cross-fertilization is documented by a multitude of powerful hybrid algorithms that were obtained by combining components from several different optimization techniques. Hereby, hybridization is not restricted to the combination of different metaheuristics but includes, for example, the combination of exact algorithms and metaheuristics. In this work we provide a survey of some of the most important lines of hybridization. The literature review is accompanied by the presentation of illustrative examples.

Edge sets: an effective evolutionary coding of spanning trees

The fundamental design choices in an evolutionary algorithm (EA) are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in EAs for network design problems directly as sets of their edges and we describe initialization, recombination, and mutation operators for this representation. The operators offer locality, heritability, and computational efficiency. Initialization and recombination depend on an underlying random spanning-tree algorithm. Three choices for this algorithm, based on the minimum spanning-tree algorithms of Prim and Kruskal and on random walks, respectively, are examined analytically and empirically. We demonstrate the usefulness of the edge-set encoding in an EA for the NP-hard degree-constrained minimum spanning-tree problem. The algorithms operators are easily extended to generate only feasible spanning trees and to incorporate local, problem-specific heuristics. Comparisons of this algorithm to others that encode candidate spanning trees via the Blob Code, with network random keys, and as strings of weights indicate the superiority of the edge-set encoding, particularly on larger instances.

Combining metaheuristics and exact algorithms in combinatorial optimization: a survey and classification

In this survey we discuss different state-of-the-art approaches of combining exact algorithms and metaheuristics to solve combinatorial optimization problems. Some of these hybrids mainly aim at providing optimal solutions in shorter time, while others primarily focus on getting better heuristic solutions. The two main categories in which we divide the approaches are collaborative versus integrative combinations. We further classify the different techniques in a hierarchical way. Altogether, the surveyed work on combinations of exact algorithms and metaheuristics documents the usefulness and strong potential of this research direction.

A unified view on hybrid metaheuristics

Manifold possibilities of hybridizing individual metaheuristics with each other and/or with algorithms from other fields exist. A large number of publications documents the benefits and great success of such hybrids. This article overviews several popular hybridization approaches and classifies them based on various characteristics. In particular with respect to low-level hybrids of different metaheuristics, a unified view based on a common pool template is described. It helps in making similarities and different key components of existing metaheuristics explicit. We then consider these key components as a toolbox for building new, effective hybrid metaheuristics. This approach of thinking seems to be superior to sticking too strongly to the philosophies and historical backgrounds behind the different metaheuristic paradigms. Finally, particularly promising possibilities of combining metaheuristics with constraint programming and integer programming techniques are highlighted.

The Multidimensional Knapsack Problem: Structure and Algorithms

We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different integer linear programming (ILP)-based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP relaxation and the original problem and then introduce a new core concept for the multidimensional knapsack problem (MKP), which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run times are also performed to compare the solutions of our approaches to the best-known solutions of another so-far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.

Models and algorithms for three-stage two-dimensional bin packing

We consider the three-stage two-dimensional bin packing problem (2BP) which occurs in real-world applications such as glass, paper, or steel cutting. We present new integer linear programming formulations: models for a restricted version and the original version of the problem are developed. Both only involve polynomial numbers of variables and constraints and effectively avoid symmetries. Those models are solved using CPLEX. Furthermore, a branch-and-price (B&P) algorithm is presented for a set covering formulation of the unrestricted problem, which corresponds to a Dantzig-Wolfe decomposition of the polynomially-sized model. We consider column generation stabilization in the B&P algorithm using dual-optimal inequalities. Fast column generation is performed by applying a hierarchy of four methods: (a) a fast greedy heuristic, (b) an evolutionary algorithm, (c) solving a restricted form of the pricing problem using CPLEX, and finally (d) solving the complete pricing problem using CPLEX. Computational experiments on standard benchmark instances document the benefits of the new approaches: The restricted version of the integer linear programming model can be used to quickly obtain near-optimal solutions. The unrestricted version is computationally more expensive. Column generation provides a strong lower bound for 3-stage 2BP. The combination of all four pricing algorithms and column generation stabilization in the proposed B&P framework yields the best results in terms of the average objective value, the average run-time, and the number of instances solved to proven optimality.

An efficient evolutionary algorithm for the degree-constrained minimum spanning tree problem

The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degree-constrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d/spl ges/2. Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(|V|) time. In addition, it is shown how problem-dependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the time-complexity. Empirical results are presented for hard problem instances with up to 500 vertices. Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.

Greedy heuristics and an evolutionary algorithm for the bounded-diameter minimum spanning tree problem

Given a connected, weighted, undirected graph G and a bound D, the bounded-diameter minimum spanning tree problem seeks a spanning tree on G of lowest weight in which no path between two vertices contains more than D edges. This problem is NP-hard for 4 < D < n - 1, where n is the number of vertices in G. An existing greedy heuristic for the problem, called OTTC, is based on Prims algorithm. OTTC usually yields poor results on instances in which the triangle inequality approximately holds; it always uses the lowest-weight edges that it can, but such edges do not in general connect the interior nodes of low-weight bounded-diameter trees. A new randomized greedy heuristic builds a bounded-diameter spanning tree from its center vertex or vertices. It chooses each next vertex at random but attaches the vertex with the lowest-weight eligible edge. This algorithm is faster than OTTC and yields substantially better solutions on Euclidean instances. An evolutionary algorithm encodes spanning trees as lists of their edges, augmented with their center vertices. It applies operators that maintain the diameter bound and always generate valid offspring trees. These operators are efficient, so the algorithm scales well to larger problem instances. On 25 Euclidean instances of up to 1000 vertices, the EA improved substantially on solutions found by the randomized greedy heuristic.

Empirical Analysis of Locality, Heritability and Heuristic Bias in Evolutionary Algorithms: A Case Study for the Multidimensional Knapsack Problem

Our main aim is to provide guidelines and practical help for the design of appropriate representations and operators for evolutionary algorithms (EAs). For this purpose, we propose techniques to obtain a better understanding of various effects in the interplay of the representation and the operators. We study six different representations and associated variation operators in the context of a steady-state evolutionary algorithm for the multidimensional knapsack problem. Four of them are indirect decoder-based techniques, and two are direct encodings combined with different initialization, repair, and local improvement strategies. The complex decoders and the local improvement and repair strategies make it practically impossible to completely analyze such EAs in a fully theoretical way. After comparing the general performance of the chosen EA variants for the multidimensional knapsack problem on two benchmark suites, we present a hands-on approach for empirically analyzing important aspects of initialization, mutation, and crossover in an isolated fashion. Static, inexpensive measurements based on randomly created solutions are performed in order to quantify and visualize specific properties with respect to heuristic bias, locality, and heritability. These tests shed light onto the complex behavior of such EAs and point out reasons for good or bad performance. In addition, the proposed measures are also examined during actual EA runs, which gives further insight into dynamic aspects of evolutionary search and verifies the validity of the isolated static measurements. All measurements are described in a general way, allowing for an easy adaption to other representations and problems.

Combining (Integer) Linear Programming Techniques and Metaheuristics for Combinatorial Optimization

Summary. Several different ways exist for approaching hard optimization problems. Mathematical programming techniques, including (integer) linear programming based methods, and metaheuristic approaches are two highly successful streams for combinatorial problems. These two have been established by different communities more or less in isolation from each other. Only over the last years a larger number of researchers recognized the advantages and huge potentials of building hybrids of mathematical programming methods and metaheuristics. In fact, many problems can be practically solved much better by exploiting synergies between these different approaches than by “pure” traditional algorithms. The crucial issue is how mathematical programming methods and metaheuristics should be combined for achieving those benefits. Many approaches have been proposed in the last few years. After giving a brief introduction to the basics of integer linear programming, this chapter surveys existing techniques for such combinations and classifies them into ten methodological categories.